\newproblem{lay:1_9_17}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.9.17}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $T(x_1,x_2,x_3,x_4)=(x_1+2x_2,0,2x_2+x_4,x_2-x_4)$. Show that $T$ is a linear transformation by finding a matrix that implements the mapping.
}{
  % Solution
	If we define $\mathbf{x}=(x_1,x_2,x_3,x_4)$, then we may define $T$ as
	\begin{center}
		$T(\mathbf{x})=\begin{pmatrix}1 & 2 & 0 & 0\\ 0 & 0 & 0 & 0\\0 & 2 & 0 & 1 \\ 0 & 1 & 0 & -1\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}$
	\end{center}
	Since this transformation is a matrix transformation of the form $T(\mathbf{x})=A\mathbf{x}$, then it is a linear transformation.
}
\useproblem{lay:1_9_17}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
